Prove limit of recursive sequence using epsilon-N

Question

I have to use the epsilon-N definition of convergence (so not monotone convergence) to prove the limit of the following sequence, and I’m really stuck on how to do so:

a_n = a_n-1 + 1/(n(n+1))

So, essentially,

a₁ = 1/2

a₂ = 1/2 + 1/6

a₃ = 1/2 + 1/6 + 1/12

a_n = 1/2 + 1/6 + 1/12 + … + 1/(n(n+1))

Intuitively, this converges to 1, so I have my candidate limit. I’m just not sure how to find N or use the epsilon-N method with a recursive sequence like this where each term is a sum including the previous terms.